Mike's Notes
Very useful.
Here is a summary generated by Google.
"Harmonic analysis of Gaussian Multiplicative Chaos (GMC) on the circle involves studying its Fourier coefficients, revealing key properties like its near-certainty of being a Rajchman measure, meaning coefficients vanish at high frequencies, and connecting to random matrix theory (Circular-β-ensembles) and spectral theory, with recent work proving exact Fourier dimensions and describing convergence laws for scaled coefficients. This field investigates how random multiplicative structures on the circle behave under Fourier transformation, revealing spectral properties and connections to number theory and random matrix models, with significant recent advances in understanding critical and subcritical phases.
Key Concepts & Findings:
- Rajchman Property: GMC on the circle almost surely becomes a Rajchman measure, meaning its Fourier coefficients 𝜇̂(𝑛) tend to zero as the frequency 𝑛 → ∞.
- Fourier Dimensions: Researchers establish precise Fourier dimensions (how fast coefficients decay) for various GMC models, confirming conjectures and linking to spectral properties.
- Connection to Random Matrices: There's a deep link between GMC on the circle and the Circular-β-Ensemble (CBE) from random matrix theory, with holomorphic analogues (HMC) also studied.
- Spectral Approach: Using random orthogonal polynomials and spectral methods allows for efficient analysis of GMC properties, including its support's Hausdorff dimension.
- Convergence Laws: Normalized Fourier coefficients are shown to converge in distribution to specific random variables (e.g., complex normal) in certain phases (L1-phase).
- Critical & Subcritical Phases: Significant focus is on subcritical 𝛾 = √2 and critical 𝛾 = √2 regimes, with new methods improving understanding, especially for the unit interval and circle.
Recent Research Directions:
- Proving exact Fourier decay rates and dimensions for various GMC models (e.g., critical GMC).
- Developing unified approaches for classical multiplicative chaos measures.
- Studying the holomorphic analogue (HMC) and its convergence properties.
- Exploring connections to number theory and large random matrices.
In essence, researchers use harmonic analysis tools (like Fourier coefficients) to understand the random, fractal-like structures of GMC on the circle, revealing underlying deterministic patterns and links to other mathematical areas."
- Gemini 3
Resources
- https://www.scientificamerican.com/article/mathematicians-crack-a-fractal-conjecture-on-chaos/
- https://arxiv.org/abs/2311.04027
- https://math.univ-lyon1.fr/~garban/papers.html
- https://arxiv.org/abs/2507.23494
- https://arxiv.org/abs/2411.13923
- https://www.scientificamerican.com/article/mathematicians-discover-prime-number-pattern-in-fractal-chaos/
- https://www.scientificamerican.com/article/fractals-chaos-video/
- https://link.springer.com/article/10.1007/s00220-015-2362-4
- https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/
- https://en.wikipedia.org/wiki/Fourier_transform
References
- Reference
Repository
- Home > Ajabbi Research > Library >
- Home > Handbook >
Last Updated
16/12/2025
Mathematicians Crack a Fractal Conjecture on Chaos
Lyndie Chiou is a scientist, a science writer and founder of ZeroDivZero, a science conference website. Her writing has also appeared in Sky & Telescope. Follow her on X @lyndie_chiou.
A type of chaos found in everything from prime numbers to turbulence can unify a pair of unrelated ideas, revealing a mysterious, deep connection that disappears without randomness.
The world may seem orderly, but randomness and chaos shape everything in the universe, from enormous galaxies all the way down to subatomic particles. Take a chilly window sheeting over with ice: even one oddly shaped snowflake can exert an influence on the final frosty pattern.
Understanding how random fluctuations can ripple out to produce global effects is what French mathematician Vincent Vargas of the University of Geneva in Switzerland set out to do more than 10 years ago. His earliest ideas for simple geometries appeared in a decade-old paper, but it wasn’t until 2023, while he was working with Christophe Garban of the University of Lyon in France, that the concept finally crystallized into what is now known as the Garban-Vargas conjecture. Now mathematicians have proved the conjecture using an insightful technique that should open the door for understanding much more complex systems.
The conjecture involves the behavior of a form of randomness found in a huge range of fields, from quantum chaos to Brownian motion to air turbulence. Mathematicians use a mathematical “measuring tape” called Gaussian multiplicative chaos, or GMC, to pick out subtle patterns hidden inside an otherwise impenetrable sea of randomness. GMC has even been used to find patterns in the prime numbers. The topic is one of the most important and fundamental ideas in probability theory today.
French mathematician Jean-Pierre Kahane is credited with first developing GMC in 1985, although his pioneering work was quickly forgotten. “I was one of the people who revived his work,” Vargas says. “I met him many times, and he said he was amazed how important the topic [had] become. Everywhere on the planet, people are working on something related to Gaussian chaos.”
Vargas first encountered the measure while studying turbulence and finance. He then came across it again in a project on conformal field theory, which is used to study patterns that remain constant as you zoom in or out. Lately he has focused on investigating its fundamental mathematical nature.
To understand GMC, imagine a turbulent fluid full of swirling eddies at many different scales. Enormous eddies randomly break apart into smaller ones, which themselves break into even smaller eddies, in a vast, nested hierarchy of randomness. GMC serves as a mathematical model that measures this kind of multiscale randomness—it captures random fluctuations that persist across every scale of the observation. Because of this, it is often referred to as a fractal measure.
Mathematicians have uncovered surprising behaviors in the types of randomness governed by GMC. For instance, events at the smallest scales can govern the entire system; the powerful tendrils of fractal structure shape chaos at every level. As a result, these systems cannot be understood by looking at averages. Instead the rules of GMC produce a universal picture that applies to every scale.
But this fascinating picture only holds up to a critical threshold. If the underlying randomness becomes too strong, the GMC measure collapses. Or, in the language of eddies, once enough randomness infuses the swirls, they become unstable, losing all their hidden order. Like ice transitioning to a liquid, this breakdown marks an important phase transition for chaos.
In 2023 Garban and Vargas introduced a new lens for studying GMC chaos. It came from a field of mathematics called harmonic analysis. Instead of looking at eddies directly, they examined the frequencies of patterns hidden in the eddies, much like analyzing a complex sound by breaking it into pure tones.
Then an idea came to them. If they could match two completely different physical descriptions—complexity and harmonics—they might learn something new. Mathematicians refer to this idea of matching unrelated physical descriptions as matching “dimensions.”
As an example, consider snowflakes falling to the ground. As the snow gently lands, two possible dimensions might be how many patterns appear in the distribution of the snowflakes and how many clumpy piles form across different scales. But is there a formula that can relate the two dimensions of patterns (harmonics) and clumpiness (correlations)?
“The key word is dimension,” Vargas says. “That’s the name of the game. You have lots of natural dimensions, but when do they coincide?”
After studying systems governed by GMC on a circle, the duo conjectured an extraordinarily elegant equation that matched a GMC system’s correlation dimension to its harmonic dimension.
Unfortunately they couldn’t prove their formula, even for a simple geometry. In 2023 they posted their conjecture to the preprint server arXiv.org, and it subsequently became a major open problem.
In 2024 mathematicians Zhaofeng Lin and Yanqi Qiu of the Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, and Mingjie Tan of Wuhan University resolved the conjecture. Their research, which was posted as a preprint to arXiv.org and has not yet been peer-reviewed, not only confirmed the formula but also revealed why it works.
Mathematically, they likened GMC to a “fair betting game,” in which the expected winnings remain constant no matter the size of the game. When applied to fractal fluctuations, this means that the system remains balanced as you zoom in and out, and each smaller scale contributes randomness in a way that conserves energy.
Mathematicians call a process that exhibits this type of fair, scale-by-scale behavior a martingale. Unlike normal betting games, however, chaos “games” are much more complex, requiring higher-dimensional martingales.
“I heard about this conjecture during an online math workshop,” Qiu says. “I had focused on martingales for my Ph.D. thesis a few years back, and I had a hunch they would be the right tool here.”
The group used its higher-dimensional martingale structure to carefully track the accumulation of randomness at every scale. And sure enough, by conserving energy, numerous tiny “fair games” combined to give the same formula for the decay that Garban and Vargas had conjectured.
Qiu and his colleagues’ proof not only settled the conjecture but also paved the way for further proofs on more complex fractal models. The roadway to a complete theory isn’t entirely free of barriers, though. Even the new method fails when randomness forces the system to its critical phase-transition point. This phase transition itself is a rich and intriguing topic with its own set of deep questions, mathematicians say. But “to go further,” Qiu says, “we need new ideas.”
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